Homomorphism from Infinite Graph to finite Graph

Question

How to prove that if there is a homomorphism from every subgraph G' of infinite graph G to a finite and fixed graph H there is also a homomorphism from G to H?

Answer 1

As noted in comments, this is trivial as stated, since $G$ is a subgraph of itself. You probably meant to ask how to prove the following statement:

If there is a homomorphism from every finite subgraph of a graph $G$ to a certain finite graph $H$, then there is also a homomorphism from $G$ to $H$.

This is a generalization of the De Bruijn–Erdős theorem which is the case $H=K_n$. Any of the usual proofs of the De Bruijn–Erdős theorem can easily be adapted to prove the more general statement. Here is a proof using the Tychonoff product theorem.

Consider $V(H)$ as a topological space with the discrete topology; then the product space $$X=\prod_{v\in V(G)}V(H)$$ is compact by Tychonoff's theorem. For each edge $e=uv\in E(G)$ define $$F_e=\{f\in X:f(u)f(v)\in E(H)\}.$$ It follows from the definition of the product topology and the fact that $H$ is finite that each $F_e$ is a closed set in $X$. Moreover, from the fact that every finite subgraph of $G$ admits a homomorphism to $H$, it follows that $\{F_e:e\in E(G)\}$ has the finite intersection property. Since $X$ is compact, it follows that the intersection $$F=\bigcap_{e\in E(G)}F_e$$ is nonempty. Plainly any element of $F$ is a homomorphism from $G$ to $H$.